Is $\nleq$ equivalent to $>$?

Question

This seems to be the case, but if so, how can we write that an inequality is not true in general.

For example, what if I want to say, $f(x,a)$ is not always less than $f(x,b)$ for all $x$ where $a$ and $b$ are constants. In other words, $f(x,a) \leq f(x,b)$ is a false statement.

Basically, I have a hypothesis that ended up being false, and I am deciding to how write the result.

Answer 1

When it comes to two real numbers $a$ and $b$: $a \not <b$ iff $a \geq b$

But, your

$f(x,a)$ is not always less than $f(x,b)$

is not equivalent to (in case that that is what you're thinking):

$f(x,a)$ is always greater or equal to $f(x,b)$

Rather, it is equivalent to:

$f(x,a)$ is sometimes greater or equal to $f(x,b)$

To be a little more formal:

I interpret your:

$f(x,a)$ is not always less than $f(x,b)$

as:

it is not the case that $f(x,a)$ is always less than $f(x,b)$

and that as:

it is not the case that for all $x$: $f(x,a)$ is less than $f(x,b)$

which is equivalent to:

for some $x$, it is not the case that $f(x,a)$ is less than $f(x,b)$

which by the basic equivalence between $\not <$ and $\geq$ is equivalent to:

for some $x$, $f(x,a)$ is greater or equal to $f(x,b)$

which you can read as:

$f(x,a)$ is sometimes greater or equal to $f(x,b)$

But this is not the same as:

for all $x$, $f(x,a)$ is greater or equal to $f(x,b)$

which you can read as:

$f(x,a)$ is always greater or equal to $f(x,b)$